A182402 Total number of digits in n-th row of a triangle formed by the positive integers.

1, 2, 3, 5, 10, 12, 14, 16, 18, 20, 22, 24, 26, 34, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 171, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset

1,2

Comments

Sequence is nonlinear at each decade transition; for example, row-5 transitions from single-digit (7) to double-digit (10) where sequence jumps (3) to (5); row-14 transitions from 2-digit (92) to 3-digit (105) where sequence jumps from (26) to (34).

The rows of nonlinearity are given by A068092. - Jon Perry, May 26 2013

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = A058183(A000217(n)) - A058183(A000217(n-1)), n >= 2. - Omar E. Pol, Jun 25 2012

Example

1; .................... (row 1 contains 1 digit)
2,   3; ............... (row 2 contains 2 digits)
4,   5,  6; ........... (row 3 contains 3 digits)
7,   8,  9, 10; ....... (row 4 contains 5 digits)
11, 12, 13, 14, 15; ... (row 5 contains 10 digits)

Mathematica

f[n_] := Length@ Flatten[ IntegerDigits[ Range[n (n - 1)/2 + 1, n (n + 1)/2]]]; Array[f, 58] (* Robert G. Wilson v, Sep 04 2013 *)

Prog

(Haskell)
a182402 n = a182402_list !! (n-1)
a182402_list = map (sum . map a055642) $ t 1 [1..] where
   t i xs = ys : t (i + 1) zs where
     (ys, zs) = splitAt i xs
-- Reinhard Zumkeller, May 26 2013
(PARI) a(n) = {my(x=n*(n-1)/2+1, y=n*(n+1)/2, nx=#Str(x), ny=#Str(y), s=0); for (i=nx, ny, if (i==nx, if (i==ny, s+=(y+1-x)*i, s+=(10^i-x)*i), if (i==ny, s+=(y+1-10^(i-1))*i, s+=i*(10^(i+1)-10^i+1)); ); ); s; } \\ Michel Marcus, Jan 26 2022
(Python)
def a(n): return len("".join(str(i) for i in range(n*(n+1)//2+1, (n+1)*(n+2)//2+1)))
print([a(n) for n in range(58)]) # Michael S. Branicky, Jan 26 2022

Crossrefs

Cf. A055642, A226029 (first differences).

Cf. A068092.

Sequence in context: A002263 A249991 A039022 * A079251 A171785 A050051

Adjacent sequences: A182399 A182400 A182401 * A182403 A182404 A182405

Keyword

nonn,base

Author

Dave Durgin, Jun 19 2012

Extensions

Better definition from Omar E. Pol, Jun 25 2012

Status

approved